Simplifying $9^{3/4}$, I get $3\sqrt[4]{9}$, but that's not the answer. Why?

Question

I am trying to simplify:

$9^\frac{3}{4}$

So this is what I did:

$9^\frac{3}{4} = \sqrt[4]{9^3}$

$\sqrt[4]{3*3*3*3*3*3}$

$3\sqrt[4]{3*3}$

$3\sqrt[4]{9}$

$3\sqrt[4]{3^2}$

I don't see how I can simplify this even more, however the answer I provided is incorrect. How can I simplify this even more?

Your answer is correct, but the fourth root of $3^2$ is the same thing as the square root of 3. That is probably what your book or instructor wants. — symplectomorphic, Dec 28 '16 at 04:21
Because $\sqrt{x}\cdot\sqrt{x}\cdot\sqrt{x}\cdot\sqrt{x}=x^2$. Hence $\sqrt[4]{x^2}=\sqrt{x}$. — symplectomorphic, Dec 28 '16 at 04:25

score 3 · Accepted Answer · answered Dec 28 '16 at 04:20

3

We know that $9=3^2$ .So, $$\sqrt [4]{9^3} =\sqrt [4]{(3^2)^3} =\sqrt [4]{3^2*3^2*3^2} =\sqrt {3*3*3*3*3*3} $$ After this you have proceeded correctly. You can simplify the last step as: $$ 3\sqrt [4]{3^2} =3\times 3^{2/4} =3\times 3^{1/2} =3\sqrt {3} $$ Hope it helps.

answered Dec 28 '16 at 04:20

Interesting! is there any way to do it without having to convert it into a fraction again? – Pablo Dec 28 '16 at 04:22
This is actually the most simplest method, @Pablo. – Dec 28 '16 at 04:27

MPW · Answer 2 · 2016-12-28T04:29:30.947

1

~~Your answer is correct, but I would leave it as $3\sqrt[4]{9}$.~~

EDIT: In the spirit of other answers, I might try this instead:

$$9^{3/4}=(3^2)^{3/4}=3^{(2)(3/4)}=3^{6/4}=3^{3/2}=3^{1+1/2}=3\cdot3^{1/2}=3\sqrt{3} $$

edited Dec 28 '16 at 04:29

answered Dec 28 '16 at 04:19

MPW

43,638

Nilabro Saha · Answer 3 · 2016-12-28T04:47:45.597

1

Simplify it this way:

$ 9^\frac{3}{4} \\ = (3^2)^\frac{3}{4}\\ = 3^\frac{2\times 3}{4}\\ = 3^\frac{3}{2}\\ = 3^{\left(1 + \frac{1}{2}\right)}\\ = 3 \times 3^\frac{1}{2}\\ = 3 \sqrt{3} $

edited Dec 28 '16 at 04:47

answered Dec 28 '16 at 04:30

Nilabro Saha

1,703

Simplifying $9^{3/4}$, I get $3\sqrt[4]{9}$, but that's not the answer. Why?

3 Answers3